Home The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
Article
Licensed
Unlicensed Requires Authentication

The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation

  • Francisco J. Mendoza EMAIL logo , Juan H. Arredondo , Salvador Sánchez-Perales , Oswaldo Flores-Medina and Edgar Torres-Teutle
Published/Copyright: April 6, 2023
Become an author with De Gruyter Brill
Author Information Explore this Subject
Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 30 Issue 3

Abstract

In the classical Fourier analysis, the representation of the double Fourier transform as the integral of f ( x , y ) exp ( − i ⟨ ( x , y ) , ( s 1 , s 2 ) ⟩ is usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions on R 2 . We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined on L p ⁢ ( R 2 ) , where 1 < p ≤ 2 , yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functions KP ⁢ ( R 2 ) which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane, Real Anal. Exchange 45 (2020), 2, 283–326]. A question arises about the dual space of the new space KP ⁢ ( R 2 ) .

Keywords: Fourier transform; bounded Hardy–Krause variation; Henstock–Pringsheim integral; Riemann–Stieltjes integral; Riemann–Lebesgue lemma
MSC 2010: 42C10; 46B07

Funding statement: This research was supported partially by CONACyT-SNI, Mexico.

Acknowledgements

The authors express their sincere gratitude to Nancy Keranen for her selfless assistance. They thank the referees for their valuable comments and suggestions which help improve this paper.

References

[1] C. R. Adams and J. A. Clarkson, Properties of functions f ⁢ ( x , y ) of bounded variation, Trans. Amer. Math. Soc. 36 (1934), no. 4, 711–730. 10.1090/S0002-9947-1934-1501762-6Search in Google Scholar

[2] D. D. Ang, K. Schmitt and L. K. Vy, A multidimensional analogue of the Denjoy–Perron–Henstock–Kurzweil integral, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), no. 3, 355–371. 10.36045/bbms/1105733252Search in Google Scholar

[3] T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, 1991. Search in Google Scholar

[4] R. G. Bartle, A Modern Theory of Integration, Grad. Stud. Math. 32, American Mathematical Society, Providence, 2001. Search in Google Scholar

[5] O. Flores-Medina, J. H. Arredondo, J. A. Escamilla-Reyna and F. J. Mendoza-Torres, On the factorization theorem for the tensor product of integrable distributions, Ann. Funct. Anal. 11 (2020), no. 1, 118–136. 10.1007/s43034-019-00026-zSearch in Google Scholar

[6] V. Fülöp and F. Móricz, Order of magnitude of multiple Fourier coefficients of functions of bounded variation, Acta Math. Hungar. 104 (2004), no. 1–2, 95–104. 10.1023/B:AMHU.0000034364.78876.afSearch in Google Scholar

[7] B. L. Ghodadra and V. Fülöp, On the convergence of double Fourier integrals of functions of bounded variation on R 2 , Studia Sci. Math. Hungar. 53 (2016), no. 3, 289–313. 10.1556/012.2016.53.3.1336Search in Google Scholar

[8] G. H. Hardy, On double Fourier series, and especially those which represent the double zeta-function with real and in commensurable parameters, Quart. J. Math. 37 (1905), 53–79. Search in Google Scholar

[9] R. Henstock, A problem in two-dimensional integration, J. Aust. Math. Soc. Ser. A 35 (1983), no. 3, 386–404. 10.1017/S1446788700027087Search in Google Scholar

[10] R. Henstock, Lectures on the Theory of Integration, Ser. Real Anal. 1, World Scientific, Singapore, 1988. 10.1142/0510Search in Google Scholar

[11] R. Henstock, The General Theory of Integration, Oxford Math. Monogr., The Clarendon Press, Oxford, 1991. 10.1093/oso/9780198535669.001.0001Search in Google Scholar

[12] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups, Grundlehren Math. Wiss. 152, Springer, New York, 1970. 10.1007/978-3-662-26755-4_3Search in Google Scholar

[13] J. Jarník and J. Kurzweil, A general form of the product integral and linear ordinary differential equations, Czechoslovak Math. J. 37(112) (1987), no. 4, 642–659. 10.21136/CMJ.1987.102191Search in Google Scholar

[14] J. Jarník, J. Kurzweil and Š. Schwabik, On Mawhin’s approach to multiple nonabsolutely convergent integral, Časopis Pěst. Mat. 108 (1983), no. 4, 356–380. 10.21136/CPM.1983.118183Search in Google Scholar

[15] Z. Kadelburg and M. Marjanović, Interchanging two limits, Enseign. Math. 8 (2005), no. 1, 15–29. Search in Google Scholar

[16] A. N. Kolmogorov and S. V. Fomin, Elementos de la teoría de funciones y del análisis funcional, Editorial MIR, Moscow, 1975. Search in Google Scholar

[17] J. Kurzweil, On Fubini theorem for general Perron integral, Czechoslovak Math. J. 23(98) (1973), 286–297. 10.21136/CMJ.1973.101167Search in Google Scholar

[18] J. Kurzweil, Nichtabsolut konvergente Integrale, Teubner Text. Math. 26, BSB B. G. Teubner, Leipzig, 1980. Search in Google Scholar

[19] P. Y. Lee and R. Výborný, Integral: An Easy Approach After Kurzweil and Henstock, Austral. Math. Soc. Lect. Ser.14, Cambridge University, Cambridge, 2000. Search in Google Scholar

[20] T. Y. Lee, Henstock–Kurzweil Integration on Euclidean Spaces, Ser. Real Anal. 12, World Scientific, Hackensack, 2011. 10.1142/7933Search in Google Scholar

[21] A. S. Leonov, Remarks on the total variation of functions of several variables and on a multidimensional analogue of Helly’s choice principle (in Russian), Mat. Zametki 63 (1998), no. 1, 69–80; translation in Math. Notes 63 (1998), no. 1-2, 61–71. Search in Google Scholar

[22] E. Liflyand, Multiple Fourier transforms and trigonometric series in line with Hardy’s variation, Nonlinear Analysis and Optimization, Contemp. Math. 659, American Mathematical Society, Providence (2016), 135–155. 10.1090/conm/659/13147Search in Google Scholar

[23] P. Muldowney and V. A. Skvortsov, An improper Riemann integral and the Henstock integral in R n (in Russian), Mat. Zametki 78 (2005), no. 2, 251–258; translation in Math. Notes 63 (2005), no. 1-2, 228–233. Search in Google Scholar

[24] J. Mawhin, Generalized multiple Perron integrals and the Green–Goursat theorem for differentiable vector fields, Czechoslovak Math. J. 31(106) (1981), no. 4, 614–632. 10.21136/CMJ.1981.101777Search in Google Scholar

[25] F. J. Mendoza Torres and M. G. Morales Macías, On the convolution theorem for the Fourier transform of B ⁢ V 0 functions, J. Class. Anal. 7 (2015), no. 1, 63–71. 10.7153/jca-07-06Search in Google Scholar

[26] F. J. Mendoza Torres, M. G. Morales Macías, J. A. Escamilla Reyna and J. H. Arredondo Ruiz, Several aspects around the Riemann-Lebesgue lemma, J. Adv. Res. Pure Math. 5 (2013), no. 3, 33–46. 10.5373/jarpm.1458.052712Search in Google Scholar

[27] M. G. Morales, J. H. Arredondo and F. J. Mendoza, An extension of some properties for the Fourier transform operator on L p ⁢ ( R ) spaces, Rev. Un. Mat. Argentina 57 (2016), no. 2, 85–94. Search in Google Scholar

[28] F. Móricz, Pointwise behavior of double Fourier series of functions of bounded variation, Monatsh. Math. 148 (2006), no. 1, 51–59. 10.1007/s00605-005-0347-7Search in Google Scholar

[29] F. Móricz, Pointwise convergence of double Fourier integrals of functions of bounded variation over R 2 , J. Math. Anal. Appl. 424 (2015), no. 2, 1530–1543. 10.1016/j.jmaa.2014.12.007Search in Google Scholar

[30] K. Ostaszewski, The space of Henstock integrable functions of two variables, Internat. J. Math. Math. Sci. 11 (1988), no. 1, 15–21. 10.1155/S0161171288000043Search in Google Scholar

[31] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Brooks/Cole Ser. Adv. Math., Brooks/Cole, Pacific Grove, 2002. Search in Google Scholar

[32] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill Ser. Higher Math., McGraw-Hill, New York, 1987. Search in Google Scholar

[33] G. E. Shilov and B. L. Gurevich, Integral, Measure and Derivative: A Unified Approach, Prentice-Hall, Englewood Cliffs, 1966. Search in Google Scholar

[34] C. Swartz, Introduction to Gauge Integrals, World Scientific, Singapore, 2001. 10.1142/4361Search in Google Scholar

[35] E. Talvila, The distributional Denjoy integral, Real Anal. Exchange 33 (2008), no. 1, 51–82. 10.14321/realanalexch.33.1.0051Search in Google Scholar

[36] E. Talvila, The continuous primitive integral in the plane, Real Anal. Exchange 45 (2020), no. 2, 283–326. 10.14321/realanalexch.45.2.0283Search in Google Scholar

[37] A. Zygmund, Trigonometric Series. Vol. I, II, 3rd ed., Cambridge Math. Libr., Cambridge University, Cambridge, 2002. Search in Google Scholar
Received: 2022-03-05
Revised: 2022-10-16
Accepted: 2022-10-27
Published Online: 2023-04-06
Published in Print: 2023-06-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
  3. On completely non-Baire unions in category bases
  4. Square-free values of n 2 + n + 1
  5. The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
  6. Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
  7. The triharmonic equation on the Heisenberg group
  8. On Busemann–Feller extensions of translation invariant density differentiation bases
  9. Multiplicative bi-skew Jordan triple derivations on prime ∗-algebra
  10. Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
  11. The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
  12. On the stochastic integral representation of Brownian functionals
  13. Generalized Bessel matrix functions
  14. Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
  15. Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
  16. 𝐿𝑝(⋅) − 𝐿𝑞(⋅) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
  17. Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
https://doi.org/10.1515/gmj-2023-2008
Keywords for this article
Fourier transform; bounded Hardy–Krause variation; Henstock–Pringsheim integral; Riemann–Stieltjes integral; Riemann–Lebesgue lemma

Articles in the same Issue

  1. Frontmatter
  2. The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
  3. On completely non-Baire unions in category bases
  4. Square-free values of n 2 + n + 1
  5. The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
  6. Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
  7. The triharmonic equation on the Heisenberg group
  8. On Busemann–Feller extensions of translation invariant density differentiation bases
  9. Multiplicative bi-skew Jordan triple derivations on prime ∗-algebra
  10. Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
  11. The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
  12. On the stochastic integral representation of Brownian functionals
  13. Generalized Bessel matrix functions
  14. Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
  15. Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
  16. 𝐿𝑝(⋅) − 𝐿𝑞(⋅) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
  17. Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 25.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2023-2008/pdf
Scroll to top button